Question

What is the factored form of the expression over the complex numbers? $121{{x}^{2}}+36{{y}^{2}}$.

Answer 1

Hint: For solving this question you should know about the factorisation of any expression. And in this type of questions, you have to know how to calculate the factors of any expression over complex numbers. And the complex numbers will decide the answer of factorisation of this expression.

Complete step-by-step solution:
In our question it is asked to make a factored form of $121{{x}^{2}}+36{{y}^{2}}$ over the complex numbers. As we know, complex numbers are the numbers which have real value and imaginary values both. And these complex numbers can be written in the form of $a+ib$, where $a$ is the real number and $ib$ is the imaginary number which does not exist at all.
For the factorisation of any expression, we always try to divide into two parts and both the parts can be the same or they can be different by the signs. But in the quantity of real values or imaginary values or any other values it is compulsory to be the same as that value. So, our expression is,
$121{{x}^{2}}+36{{y}^{2}}$
We can write it as,
${{\left( 11x \right)}^{2}}+{{\left( 6y \right)}^{2}}$
Since we know that ${{A}^{2}}-{{B}^{2}}=\left( A+B \right)\left( A-B \right)$ and for the factorisation we have to make this as a form of $a+ib$. So, we can write it as,
$121{{x}^{2}}+36{{y}^{2}}={{\left( 11x \right)}^{2}}+{{\left( 6y \right)}^{2}}$
And it can be written as,
$121{{x}^{2}}+36{{y}^{2}}={{\left( 11x \right)}^{2}}-\left( -1 \right){{\left( 6y \right)}^{2}}$
We know that ${{i}^{2}}=-1$
So, we can write it as,
$\begin{align}
  & {{\left( 11x \right)}^{2}}-{{i}^{2}}{{\left( 6y \right)}^{2}} \\
 & \Rightarrow {{\left( 11x \right)}^{2}}-{{\left( 6iy \right)}^{2}} \\
 & \because {{A}^{2}}-{{B}^{2}}=\left( A+B \right)\left( A-B \right) \\
 & \Rightarrow \left( 11x+6iy \right)\left( 11x-6iy \right) \\
\end{align}$
So, the factored form of the expression over the complex number is $\left( 11x+6iy \right)\left( 11x-6iy \right)$.

Note: For calculating the factored form we can use the factorisation methods but if we have been asked to make the factors in the complex number form, then it is always fixed that we have to make a real number and an imaginary number which is done by iota $\left( i \right)$.